If this technique fails, PÃ³lya advises: "If you can't solve a problem, then there is an easier problem you can solve: find it." Or: "If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?"

First principle: Understand the problem

"Understand the problem" is often neglected as being obvious and is not even mentioned in many mathematics classes. Yet students are often stymied in their efforts to solve it, simply because they don't understand it fully, or even in part. In order to remedy this oversight, PÃ³lya taught teachers how to prompt each student with appropriate questions, depending on the situation, such as:

What are you asked to find or show?

Can you restate the problem in your own words?

Can you think of a picture or a diagram that might help you understand the problem?

Is there enough information to enable you to find a solution?

Do you understand all the words used in stating the problem?

Do you need to ask a question to get the answer?

The teacher is to select the question with the appropriate level of difficulty for each student to ascertain if each student understands at their own level, moving up or down the list to prompt each student, until each one can respond with something constructive.

Second principle: Devise a plan

PÃ³lya mentions that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is included:

Guess and check

Make an orderly list

Eliminate possibilities

Use symmetry

Consider special cases

Use direct reasoning

Solve an equation

Also suggested:

Look for a pattern

Draw a picture

Solve a simpler problem

Use a model

Work backward

Use a formula

Be creative

Use your head/noggin

Third principle: Carry out the plan

This step is usually easier than devising the plan. In general, all you need is care and patience, given that you have the necessary skills. Persist with the plan that you have chosen. If it continues not to work discard it and choose another. Don't be misled; this is how mathematics is done, even by professionals.

Fourth principle: Review/extend

PÃ³lya mentions that much can be gained by taking the time to reflect and look back at what you have done, what worked and what didn't. Doing this will enable you to predict what strategy to use to solve future problems, if these relate to the original problem.

The book contains a dictionary-style set of heuristics, many of which have to do with generating a more accessible problem. For example:

The technique "have I used everything" is perhaps most applicable to formal educational examinations (e.g., n men digging m ditches) problems.

The book has achieved "classic" status because of its considerable influence (see the next section).

Russian inventor Genrich Altshuller developed an elaborate set of methods for problem solving known as TRIZ, which in many aspects reproduces or parallels PÃ³lya's work.

From Yahoo Answers

Question:1.x= log^7(30)
2.f^(x^2-3) = 71
How do you do them right? Heres what i did, and i was wrong
log7+log30
.845 + 1.47= 2.3222
and for the second i did
log 72 divided by log 5 = xsquared - 3
which was 2.6572 = x squared - 3
then add 3 to get 5.6571 = x squared
then do the square root of 5.65.. which is 2.3784
but these are wrong can anyone tell me why? Q2 is
5^(x^2-3) = 72 oh nvm 2 was right

Answers:I don't understand the way you've typed the question.
The symbol ^ means "exponent", which doesn't make sense here except where you put x^2.
A guess: Does number 1 mean :"log of 30 with base 7"?
If so, we can write it here as log(7) 30 or log(base 7) 30
or log 30 to base 7.
If it's that, calculate it as (log 30)/()log 7)
=1.747869697
While using Excel for that calculation just now, I notice they use
log(30,7) to mean log of 30 with base 7. Which makes me think that maybe q.1 is log of 7 in base 30, which is the reciprocal of the answer I've found.
Q.2 What you have done would give a correct solution to the equation
5^(x^2-3) = 72
I didn't work through to check your figures, but the process is correct and the numbers look reasonable. Is that the equation you had to solve? I notice there's a 71in the equation you typed for us, but I don't think the answer would be very different whether it was 71 or 72. So tell us, what was the equation really?

Question:the problem is: 16^2x-5= 2^7
^ = the power of.
do you know how to solve this?

Answers:16^(2x-5) = 2^7
(2^4)^(2x-5) = 2^7
2^(8x-20) = 2^7
because the bases are the same you can set the powers equal to each other..
8x-20 = 7
8x = 27
x = 27/8

Question:4e^-2x=17
I don't understand how to solve it since it has e in it. I know how to do the other ones like 7^6x=12. but that doesn't have e. So could you please explain to me how to work this problem? I did try, and I got -1.0219. But the book says its -.723. So I don't know what I did wrong.. thanks! (:

Question:Solve the exponential equation algebraically. Round results to 3 decimal places.
40/(3-1e^-0.0001x) = 1000
The answer is: - 1085.189
The question I am asking is how do i work the problem to get the answer -1085.189
Please show me how to get the answer
Thanks

Solving Exponential Equations :Just a video showing how to solve for ' x ' in some cases, if ' x ' happens to be an exponent. A few different examples. For more free math videos, visit: PatrickJMT.com

Exponents & Logs: Solve Exponential Equations :www.mindbites.com This 76 minute exponents & logarithms lesson focuses on solving equations with variables as exponents, for example 9^(2x+1) = 81^4 times 27^x: This lesson will show you how to solve exponential equations: - with the same bases - with different bases - with two solutions - using a system of linear equations - by factoring Thislesson contains explanations of the concepts and 35 example questions with step by step solutions plus 6 interactive review questions with solutions. Lessons that will help you with the fundamentals of this lesson include: - 115 The 5 Basic Exponent Laws (www.mindbites.com - 165 The Zero Negative & Rational Exponent (www.mindbites.com - 205 Solving Systems of Linear Equations (www.mindbites.com - 230 Solving Quadratic Equations by Factoring (www.mindbites.com